Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

Abstract

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian H : R n → R in the regime of low temperature ε. We proof the Eyring-Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator L = ε∆ − ∇H · ∇ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302-351] and of the mean-difference estimate introduced by Chafaï and Malrieu [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 72-96]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in ε. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.

Some recent developments in functional inequalities

“…Surprisingly, it turns out that these estimates on the LSI constant do not coincide with the Eyring-Cramer formula. Additionally, we expect that the estimates on LSI constant obtained by this approach are sharp (for details see [MS12]). …”

“…Let us only consider an one-dimensional Hamiltonian H, even if the results holds in any dimension. Additionally for this short note, we will not discuss some standard growth conditions and nondegeneracy conditions on the Hamiltonian H. For precise statements, we refer the reader to the preprint [MS12] of both authors. With these simplifications, the Eyring-Kramers formula becomes:…”

“…This new approach also allows to derive sharp estimates on the log-Sobolev constant. For details we refer the reader to the preprint [MS12] of both authors.…”

“…Section 5 contains the contribution by Georg Menz and André Schlichting based on [MS12]. The question is to give a precise asymptotic for the Log-Sobolev or Poincaré constants of a Gibbs measure µ T (dx) = 1 Z T e −H(x)/T dx when T → 0 (low temperature).…”

Full Metastable Asymptotic of the Fisher Information

“…The Fisher information may enter the description and analysis of a Statistical Mechanics model in various ways: through functional inequalities, prominently the Log-Sobolev and HWI inequality (see e.g. [1] and references therein and in particular [18] for the β → ∞ regime considered here); as rate functional in Large deviation principles describing fluctuations of the empirical occupation measure of (1.3) when t → ∞, [5]; or also as a tool to give an alternative construction of the dynamics (1.2) as gradient flow with respect to the relative entropy functional [12] (see also the discussion at the end of this introduction).…”